schemata, intuition, and problem solving sara hershkovitz center for educational technology israel...

Schemata, Intuition,

andProblem Solving

Sara HershkovitzCenter for Educational Technology ISRAEL

SEMT09 - International SymposiumElementary Mathematics Teaching Prague, Czech Republic, August

20091

Kant (1724 – 1804)

The link between perceiving real world objects and categories of pure understanding

Schema

2

schema of substance,

schema of cause,

schema of community,

schema of possibility,

schema of reality,

schema of necessity,

schema of quality,

schema of relation,

schema of modality and its categories.

Schema

3

Kant distinguished:

Kant “A schema is constructed according to the necessary conditions of the unity of reason – the schema of a thing in general, which is useful towards the production of the highest degree of systematic unity in the empirical exercise of reason ….

…it merely indicates how, under the guidance of the idea, we ought to investigate the constitution and the relations of objects in the world of experience.”

Schema

4

Schemata:

"all possible objects following the arrangements of the categories"

Kant

Schema

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Category:A category is an attribute, property, quality, or characteristic that can be predicated of a thing.

Kant called them "ontological predicates.”

Aristotle claimed that the following ten predicates or categories could be asserted of anything in general: substance, quantity, quality, relation, action, affection (passivity), place, time (date), position, and state.

KantSchema

6

Piaget (1896 -1980):

A schema of an action consists of those aspects which arerepeatable, transposable, or generalisable (1980).

Schemata develop by two mechanisms: assimilation and accommodation. .

Schema

7

“The building blocks of cognition“: Rumelhart discussed schemata while taking into account different notions:

frames, scripts, retrieving information, organizing actions, allocating sources, and guiding the flow of processingfunctional relationships characteristic of an object. (1980)

Rumelhart

Schema

8

Schemata:Data structures for representing the generic concepts are stored in memory. generalized concepts underlying objects, situation, events, sequences of

events, action and sequences

Schemata represent the stereotypes of concepts.

Schemata are like models of the outside world. (1985)

Rumelhart and Norman

Schema

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To process information with the use of

schemata is to determine which model best

fits the incoming information. (1985)

Rumelhart and Norman

Schema

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Semantic nets expressing relations;

Scripts of behavior;

Schema:A mental representation of some aspect of the world;

Schank & Abelson (1977)

Anderson (1980)

Howard (1987)

Schema

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Schemata have variables;

Schemata can be embedded, one within another;

Schemata represent knowledge at all levels of abstraction; Schemata represent knowledge rather than definitions;

Schemata are active recognition devices.

Schema

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Features:

Howard (1987) following Rumelhart (1980):

A schema is like a sorting device;(allows to determine that some stimuli are instantiations and others are not).

A schema is like a play; (the parts of a play relate to each other as specified by scripts, performance…)

A schema is like a filter.(it allows us some information in but not all)

Schema

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Uses of Schemata (Howard 1987):

Perception: Providing means of recognising patterns, analyzing and interpreting new data.

Comprehension:To understand something is to assimilate it to something we know.

Schema

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“The procedure is actually quite simple. First you arrange things into different groups.

It is important not to overdo things. That is, it is better to do too few things at once than too many.

In the short run, this may not seem important but complications can easily arise. A mistake can be expensive as well.

Example: Bransford & Johnson (1973)

Schema

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At first the whole procedure will seem complicated. Soon, however, it will become just another facet of life.

After the procedure is completed one arranges the materials into different groups again. Then they can be put in their appropriate places. Eventually they will be used once more and the whole cycle will then have to be repeated”.

Example: Bransford and Johnson (1973)

Schema

16

?????

Example: Bransford and Johnson (1973)

Schema

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Reasons for failing to comprehend: (Howard 1987, Rumelhart 1980):

Not knowing appropriate schema(and cannot readily construct one)

Not comprehending the appropriate given clues to elicit a schema (even if a schema is known)

Applying a different schema to given stimuli

Schema

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Nesher (1986)

Schema is a strategy of solving a certain class of problems.

A plan for action.

Schema

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“A kind of condensed, simplified representation of a class of objects or events”

Schema

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Fischbein (1920 - 1998):

“Adaptive behavior of an organism … achieved by assimilation and accommodation”

Fischbein (1999):

A schema is a program whichenables the individual to:

Schema

Related terms:Frame; Script; Framework;Mental structure

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)a (record, process, control, and mentally integrate information ,

(b) react meaningfully and efficiently to environmental stimuli.

Knowledge Schema

KnowledgeSchema

How many cubes are there?

John had 6 marbles.He lost 2 marbles.How many marblesdoes John have now?

Counting

22

Knowledge Schema

KnowledgeSchema

Class inclusion

There are 7 fruits on the plate. 3 of them are apples and the others are peaches.How many peaches are there on the plate?

Counting ?

23

Knowledge Schema KnowledgeSchema

Part – Part – WholeIf A+B=C thenC-A=B & C-B=A

Dan had some marbles.He found 5 more marbles.Now he has 8 marbles.How many marbles did he have to start with?

Class inclusion ?

24

Knowledge Schema

KnowledgeSchema

Part – Part – WholeReversibility

Dan had some marbles.He lost 5 of them.Now he has 3 marbles.

How many marbles did he have to start with?

Part – Part – Whole ?

25

Knowledge Schema

KnowledgeSchema

Part – Part – Whole &

Infinity

Is the set of whole numbers equivalent to the set of even

numbers?

Part – Part – Wholereversibility ?

26

Knowledge Schema

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Counting

Class inclusion

Part-Part-Whole

Reversibility

Infinity

schema

Open Ended Problem

Look at the following numbers:

23, 20, 15, 25,

which number does not belong? Why?

28

schemaLook at the following numbers:

23, 20, 15, 25,which number does not belong? Why?

Solution no. 1

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schemaLook at the following numbers: 23, 20, 15, 25,


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Solution no. 2

schemaLook at the following numbers:

23, 20, 15, 25,which number does not belong? Why?

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15 - It is in the 2nd ten and the rest are in the 3rd ten

20 - The only “round” number

- This number has more factors

23 - Not a multiple of 5

- The only prime number

25 - The sum of its digits is the largest

- The only square number

Solution no. 3

schema

The more schemata – the more solutions

Look at the following numbers: 23, 20, 15, 25,


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Schema is the means by which similar experiences are assimilated and aggregated to a whole.

To summarize:

Schema

33

Schema links together many different kinds of knowledge

Schema enables to store and generalize ideas

IntuitionDescartes (1596 -1650):Intuition occurs either after or concomitantly with analysis.

Methods consist of a set of rules or procedures for using the natural capacities and operations of the mind correctly.

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Intuition - deduction

IntuitionOlscamp (2001):

analysis-intuition-deduction

Plato: related to intuition ascontinuing analysis and preceding synthesis

35

intuition

Kant:

A way in which objectsare directly grasped.

Intuition remains related tosensorial knowledge (intellectual intuition – does not exist)

36

intuition

Fischbein:

Intuitive understanding *

Formal understanding

Procedural understanding

* Related terms:Insight; Common sense; Interpretation; Inspiration; Naïve reasoningPiaget: self-evidence

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intuition

Fischbein: Intuitive understanding

Direct, self-evident

Intrinsic certainty

Coerciveness

Extrapolativeness

Globality

Affirmatory intuitions Anticipatory intuitions

38

Intuition

the effect of compression if

a structural schema lies behind this cognition

intuition

Fischbein: Intuitive understanding

Direct, self-evident

Intrinsic certainty

Coerciveness

Extrapolativeness

Globality

Affirmatory intuitionsAnticipatory intuitions

Grasping the problem

Distinguishing between the given

information and the question

Searching for strategies

Finding a schema for solving

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intuition

Example:

Solving without having a specific schema….

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Divide 21 matches into two groups so that one group will be twice as big as the other group.

Schema - intuitionSchemaIntuition

Behavior & Cognitive ability

Cognitive ability

Plan for action Appears unplanned

GlobalBuilt of components

Develops; Can be adapted Direct, Self-evident

Assimilation & Accommodation Intrinsic certainty

Analytical & Logical Global & Extrapolative

FlexibleExamined & Adjusted

CoerciveImmediate & Self-evident

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Does using schemata promote problem solving ?

1.Additive Problems2.Two-step Problems

Is it possible to teach using schemata?

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SchemaExamplesMathematicalcompetence

Simple additive problems (Nesher 1982)

Building setof objects and counting

1. There are 5 apples and 2 bananas in the bag. How many pieces of fruit are there in the bag?

2 . There were 5 apples in the bag. Dan took 2 apples out of the bag. How many apples are there in the bag?

1,2,3,4,5,

1,2,3,

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6,7

4,5

Simple additive schema

part part

whole

+

44


5 apples 2 bananas

?fruit

+

?fruit

5 apples 2 bananas

5 + 2 = 7

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1. There are 5 apples and 2 bananas in the bag. How many pieces of fruit are there in the bag?


2 apples ?apples

5 apples

+

5 apples

5 – 2 = 3

2 apples ?apples

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2. There were 5 apples in the bag. Dan took 2 apples out of the bag. How many apples are there in the bag?


Simple additive problems

Part-Part-Whole

Change

3. Roni had 3 marbles. Then Tom gave him some more marbles. Now Roni has 8 marbles. How many marbles did Tom give Roni?

3 + 8 =11

3 + ? =8(3 + 5 =8)

8 - 3 =5

X

V

V

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Part-Part-Whole

Reversibility

4. Tom had some marbles. He found 5 more

marbles. Now he has 8 marbles. How many marbles did

he start with ?

5 + 8 =13

? + 5 =8 (3 + 5 =8)

8 - 5 =3

X

V

V

Simple additive problems

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SchemaProblemWith SchemaWithout Schema

PrePostPrePost

Building sets &Counting

Combine 173.410077.8100

Change 18795.7100100

Change 28791.377.888.9

Part –Part-Whole &Change

Combine 247.882.633.344.4

Change 343.591.355.666.7

Change 460.98744.455.6

Part-Part-Whole &Reversibility

Change 560.973.922.233.3

Change 639.152.233.333.3

Ogonovski & Nesher (2009)49


PrePostPrePost


Combine 173.410077.8100

Change 18795.7100100

Change 28791.377.888.9


Combine 247.882.633.344.4

Change 343.591.355.666.7

Change 460.98744.455.6


Change 560.973.922.233.3

Change 639.152.233.333.3


The Building Blocks of word problems:

part part

whole

+factor factor

product

X

Additive Schema

Multiplicative Schema

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Two-step Word Problems

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Hierarchical Schema

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6 plates 5 oranges on each plate

?oranges

X

?oranges 15 bananas

?fruits

+

Dina has 15 bananas

and 6 plates with 5

oranges on each.

How many pieces of fruit

does Dina have?

6 X 5 +15 = 4555

?plates 5 oranges on each plate

?oranges

X

?oranges 15 bananas

45 fruits

+

Dina has 45 pieces of

fruit. 15 of them are

bananas, and the rest

are oranges. She put the

oranges on plates, 5

oranges on each plate.

How many plates are

there?

(45 – 15) : 5 = 656

6 plates 5 oranges on each plate

?oranges

X

?oranges ?bananas

45 fruits

+

Dina has 45 pieces of

fruit. She put them in 6

plates on each of which

there are 5 oranges. The

rest of the fruit are

bananas. How many

bananas does Dina

have?

45 – 6 X 5 = 1557

Shared Part Schema

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Shared Whole Schema

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Hierarchical Shared Whole

Shared Part Schema

Hershkovitz & Nesher (1992; 1994; 1996; 1998)60

Learning with or without schemata

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SPA – Schema for Problem AnalysisAP – Algebraic Proposer – J. Schwartz

Population:Two groups of 6th gradersTwice a week during 4 months

2 easy problems 2 difficult problems

StudentsProgramEasy problems

Difficult Problems

AllSPA1.781.74

AP1.720.97

Low achievers

SPA1.671.56

AP1.400.53

High achievers

SPA1.861.86

AP2.001.35

Learning with or without schematagraded 0-2

62

SPA – Schema for Problem AnalysisAP – Algebraic Proposer


Difficult Problems

AllSPA1.781.74

AP1.720.97

Low achievers

SPA1.671.56AP1.400.53

High achievers

SPA1.861.86

AP2.001.35

Learning with or without SchemaLow achievers

63


Difficult Problems

AllSPA1.781.74

AP1.720.97

Low achievers

SPA1.671.56AP1.400.53

High achievers

SPA1.861.86AP2.001.35

Learning with or without schemataDifficult Problems

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Pathway between Text and Solution of Word Problems

Hershkovitz & Nesher (2001)

Word Problem

Solution

Understanding the given text

Constructing a representation Finding an appropriate schemaApplying the schema to given informationConstructing the math model

Solving the problem

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Word problems:3 two-step word problems Typical of problems taught in math classes

Population:49 fifth and sixth grade Israeli students were individually interviewed in a single 45-minute session

Hershkovitz & Nesher (2001):

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Word Problems:

Problem No. 1: I have a book of 320 pages. I already read 80 pages. How many days are needed to finish reading the book if I read 60 pages each day? Problem No. 2: In the morning the flower seller distributed the roses equally into 6 vases. How many roses did he place in each vase if during the day he sold 120 roses and at the end of the day 60 roses were left? Problem No. 3: Lunch boxes were prepared for all participants. Each lunch box had 5 pieces of fruit of which2 were apples and the rest were plums. In preparing the lunch boxes 240 plums were used. How many participants received lunch boxes?

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Interviews:

Every student was asked, for each of the three problems, to: Read aloud the text (original text) of the word problem Retell it (first retelling) Solve it. After solving the problem, the student was asked to retell the story again (second retelling).

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Analysis of:

Deviation from the original text in the retelling

(a) Changing the wording without changing the schema, usually by adding details to the description of the situation (episode), which were taken from general world knowledge, and were not mentioned in the text.

(b) Changing the order of the text.

(c ) Retelling the original text exactly.

(d) Changing the schema of the text Changes were made in the text to fit the erroneous solution

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Examples: (a) Changing the wording without changing the schema:

Original problem no. 2Yael retold problem no. 2

In the morning the flower seller distributed roses equally into 6 vases. How many roses did he place in each vase if during the day he sold 120 roses and at the end of the day 60 roses were left?

There was a seller. He received roses and equally distributed them in vases. During the day a lot of people arrived and bought a lot of roses. Then he found

out that 60 roses were left?

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Examples: (b) Changing the order of the text:

Original problem no. 2Michal retold problem no. 2

In the morning the flower seller distributed roses equally into 6 vases. How many roses did he place in each vase if during the day he sold 120 roses and at the end of the day 60 roses were left?

A flower seller sold 120 flowers and 60 flowers were left.

The flowers (those sold and those left) were in 6 vases. How many flowers were

there in each vase?

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Using both categories:)a (Changing the wording without changing the schema

)b (Changing the order of the text

Original problem no. 3

Lunch boxes were prepared for all participants.

Each lunch box had 5 pieces of fruit of which

2 were apples and the rest were plums.

In preparing the lunch boxes 240 plums were used.

How many participants received lunch boxes?

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Johnny's repetition was:

A member of the entertainment committee, or somebody else, I don’t know exactly who, prepared the lunch boxes for the trip the committee organized. In each lunch box they put 5 pieces of fruit of which there were 2 apples and 3 plums. 240 plums were needed to prepare all the lunch boxes. How many children got lunch boxes?While solving the problem he wrote 2 math expressions as follows: “3 + 2 = 5 and 240 : 3 = 80” plums apples fruitsHe summarized “80 children will get lunch boxes”.He continued and said: “Now I can find out how many apples were needed as well (80X2=160).”73

)d (Changing the schema

Shay retold problem no.3

Lunch boxes were prepared. There were 5 fruits in each lunch box, of which 3 were plums and 2 were

apples . Shay continued to speak aloud while solving:

240:5=48 and said :“240 are all the fruit. Each child received 5 fruits.”

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Findings

ProblemCorrect SolutionsIncorrectSolutions

Not * Included

(a))b()c()d(

143%29%8%6%14%

235%29%-38%4%26%6%

326%29%-33%8%33%4%

(a) Changing the wording without changing the schema(b) Changing the order(c) Retelling exactly(d) Changing the text into different schema.

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1. Almost all students who correctly solved the problems elaborated to some extent: Some

by using their world knowledge, while others began with the mathematical solution.

2. All students who failed to solve the problems changed the text into another schema, usually to a simpler one. These changes related to changing the text so that it described different mathematical structures.

The second retelling (after solving the problems) was consistent with the already incorrectly solved problem.

76

Anderson et al. (1983) claim that:

“The content schema embodies the reader’s

existing knowledge of real and imaginary

worlds.

What the reader already believes about the

topic helps to structure the interpretation of

new messages about the topic”.

77

We found that all students in our sample, who

constructed a richer text by adding detailed

information, did so because it was useful for

them in order to construct a complete

understanding of the text, find the appropriate

schema, and then solve the problems correctly.

78

Non Routine Problems (N.R.P)

79

Schema ?? Intuition ??

N.R.P – Sharing Pizza

80

There were 40 children in the summer camp. For a dinner some large pizzas and some small pizzas were ordered.Each large pizza was divide equally among the children, and each small pizza was divided equally between the children.

How many pizzas were ordered?Offer some possibilities.

N.R.P – Sharing Pizza

N.R.POpen-ended-Problem

Intuitively - general world knowledge (direct, coercive, global)

Mathematical schemata - sharing, dividing, fraction? (only after the teacher’s example)

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http://ofek.cet.ac.il/units/en/math/Unit97/act1.aspx

Final Notes

1. Intuitions and schemata are two complementary factors needed for problem solving in mathematics.

2. Descartes presented the process of "analysis-intuition-deduction" as a way to achieve certain knowledge.

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Final Notes

3. Feferman (2000) stressed:

The ubiquity of intuition in the common experience of teaching and learning mathematics … is essential for motivation of notions and results and to guide one's conceptions via tacit or explicit analogies in transfer from familiar ground to unfamiliar terrain ... intuition is necessary for the understanding of mathematics. 84

4. The more schemata a person acquires, the more intuition he has.

(Fischbein 1999)5. The educational challenge is to enable children to develop rich repertoire of mathematical schemata leading to more intuitions for solving mathematical problems.

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Intuition: ???

Analysis-intuition-synthesis;

Mathematical Schemata: Factorization, division with remainder

N.R.P – Birthday Cake

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N.R.P – Birthday Cake’s solution

Solution No.Length of pattern

11

22

33

44

55

66

710

812

915

1020

1130

126087


Solution No.Length of pattern Remainder

2217 : 2 = 8) R 1(

3317 : 3 = 5) R 2(

4417 : 4 = 4) R 1(

5517 : 5 = 3) R 2(

6617 : 6 = 2) R 5(

71017 : 10 = 1) R 7(

81217 : 12 = 1) R 5(

91517 : 15 = 1) R 2(

102017 : 20 = 0) R 17(

113017 : 30 = 0) R 17(

88


Ron 3rd grade High Achiever

Ron: Do I have to use all the candies?

Teacher: No, you can use the same color more than once

in the same pattern. You have to choose a pattern so that

the 17th candy will be purple.

Ron placed 5 different candies in this order: blue,

purple, orange, green, and red, and duplicated them.

T: How did you know?

.89


R: 17 divided by 5 gives the pattern 3 times and the

remainder is 2, so the purple has to be the second.

T: Great. Do you think there are more solutions?

R: Thought for a while… I'll try with 4 candies.

T: And?

R: The purple will be the first.

T: How?90


R: 17 divided by 4 gives 4, remainder 1. The

remainder is 1, so this is the purple candy.

T: Do you have an idea for "the rule" of the game?

Ron tried patterns with 3 candies and with 2

candies.

R: The place of the purple candy is the remainder of

the division exercise.

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Avivah (A teacher) Generalization: The number of options is 529 since the largest pattern possible is 30.

I knew that from the requirement of the task which requires to fill 60 candies in a pattern.The pattern is actually all the factors of the number since they must repeat.The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The factor 60 is not interesting, since it will not make a repeating pattern,

92

Avivah (A teacher)

therefore I stay with the factor 30 as the largest pattern. In this pattern I need to fill only 29 candies, since the 30th candy (in the 17th place) is filled and fixed according to the problem's requirement.

For each of the remaining 29 candies I have to choose among 5 colors, therefore this situation includes all the other options and gives the answer I wrote above.

93

Nethanel (6th grade)

Documentation of the solution process by Aviva First trial: puts purple. Smiles. "I'm sure it will work". SuccessSecond trial: puts purple-blue. "The same". SuccessNethanel says: "Each color instead of the blue will work. Each sequence of 2 , as long as the purple is first”.Third trial: puts red-purple-blue. SuccessNethanel says: "6 will also work".Continues and says: "9 will also work".

Fourth trial: puts 9 No success

94


Nethanel says: "Wait, why does 6 work and 9 doesn't work?" Waits for a minute. Looks disturbed.Aha! I got it! 60:3, 60:6, 60 is not divisible by 9. So 4 would also work.I ask: "Where will the purple be?"Nethanel answers: second. No, wait, not second. When you put five it will be second, no matter what the others will be.I insist: “So with 4? Where is the purple?"Nethanel answers: "Same as with 2. First."Continues: "So it can also be 10, as long as the second and the seventh will be purple."

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I ask: "Second and seventh? You must have both?"Nethanel answers: “Yes. No. Wait." Checks the options with the computer.Second and seventh SuccessSecond only No SuccessSeventh only SuccessSays: “Seventh is enough".I ask: "So, does it end?"Nethanel: "I don't know. I think there's no end."I ask: "Why?"

96


Nethanel: "I don't know."I ask: "What else can you do?"Nethanel: "Every number that 60 is divisible by, and you have to check where the purple will be."I ask: "So there is an end?"Nethanel: "There is an end to the numbers that 60 is divisible by, but I also have the 17th candy and I also have five colors. I don't know."

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Thank [email protected]

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schemata, intuition, and problem solving sara hershkovitz center for educational technology israel...

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